a generalization of the widder–arendt theorem - School of Computer

A GENERALIZATION OF THE WIDDER–ARENDT THEOREM WOJCIECH CHOJNACKI Wydzial Matematyki, Uniwersytet Kardynala Stefana Wyszy´ nskiego, ul. Dewajtis 5, 01-815 Warszawa, Poland, and Department of Computer Science University of Adelaide, Adelaide, SA 5005, Australia E-mail: [email protected]; [email protected]

Abstract We establish a generalization of the Widder–Arendt theorem from Laplace transform theory. Given a Banach space E, a non-negative Borel measure m on the set R+ of all non-negative numbers, and an element ω of R ∪ {−∞} such that −λ is m-integrable for all λ > ω, where −λ is defined by −λ (t) = exp(−λt) for all t ∈ R+ , our generalization gives an intrinsic description of functions r : (ω, ∞) → E that can be represented as r(λ) = T (−λ ) for some bounded linear operator T : L1 (R+ , m) → E and all λ > ω; here L1 (R+ , m) denotes the Lebesgue space based on m. We use this result to characterize pseudo-resolvents with values in a Banach algebra, satisfying a growth condition of Hille–Yosida type. Keywords: Laplace–Stieltjes transform, weighted convolution algebra, representation, pseudo-resolvent, one-parameter semigroup AMS 2000 Mathematics subject classification: Primary 44A10; 47A10 Secondary 43A20; 46B22; 46G10; 46J25; 47D06

1. Introduction Let F be either the field R of real numbers or the field C of complex numbers. Hereafter all vector spaces will be assumed to be over F. Any particular choice of the ground field F will be inessential for the validity of results. Let R+ be the set of all non-negative numbers and let R•+ be the set of all positive numbers. A weight function on R+ is a Lebesgue measurable function Ω : R+ → R•+ such that Ω (s + t) ≤ Ω (s)Ω (t)

(1.1)

for all s, t ∈ R+ . A function satisfying (1.1) is said to be submultiplicative. Given a weight function Ω , denote by L1 (R+ , Ω ) the set of equivalence classes of F-valued Lebesgue measurable functions f on R+ such that Z kf k1,Ω = |f (t)|Ω (t) dt < ∞, R+

where two functions are equivalent if they are equal almost everywhere. With addition and scalar multiplication derived from the addition and scalar multiplication of functions, and with the norm k · k1,Ω , L1 (R+ , Ω ) is a Banach space. Moreover, L1 (R+ , Ω ) is a 1

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W. Chojnacki

Banach algebra, a weighted convolution algebra, when multiplication is defined to be the convolution Z t (f ∗ g)(t) = f (t − s)g(s) ds (almost all t ∈ R+ ). 0

Let Z+ be the set of all non-negative integers. For each λ ∈ R, denote by λ the function λ (t) = eλt (t ∈ R+ ). We shall employ the notation f ∗n = f ∗ · · · ∗ f . | {z } n times

For each k ∈ Z+ and each λ ∈ R, set ∗(k+1)

αk,λ = −λ

,

(1.2)

βk,λ = k! αk,λ ,

(1.3)

k

γk,λ = λ αk,λ ;

(1.4)

more explicitly, αk,λ (t) =

tk −λt e , k!

βk,λ (t) = tk e−λt ,

γk,λ (t) =

(λt)k −λt e k!

for all t ∈ R+ . Let Ω be a weight function on R+ . A bound for Ω is an element ω of {−∞} ∪ R such that −µ ∈ L1 (R+ , Ω ) for each µ > ω. If ω is a bound for Ω , then αk,µ ∈ L1 (R+ , Ω ) (and automatically βk,µ ∈ L1 (R+ , Ω ) and γk,µ ∈ L1 (R+ , Ω )) for all k ∈ Z+ and all µ > ω. Let E be a Banach space. Given an open subset U of R, let C ∞ (U, E) be the space of E-valued functions on U infinitely differentiable in the norm topology of E. Let Ω be a weight function on R+ with bound ω. For r ∈ C ∞ ((ω, ∞), E), set krkW,Ω,ω = sup{kr(k) (λ)k/ kβk,λ k1,Ω | k ∈ Z+ , λ ∈ (ω, ∞)} ∞ and define the Widder space CW ((ω, ∞), E; Ω ) by ∞ CW ((ω, ∞), E; Ω ) = {r ∈ C ∞ ((ω, ∞), E) | krkW,Ω,ω < ∞}. ∞ Equipped with the norm k · kW,Ω,ω , CW ((ω, ∞), E; Ω ) is a Banach space. • A weight function Ω : R+ → R+ will be termed an (α, ω)-weight function if

(i) Ω is continuous and satisfies Ω (0) = 1; (ii) there exist α ∈ R+ and ω ∈ R such that Ω (t)t−α e−ωt tends to a positive number as t → ∞. The numbers α and ω in condition (ii) are uniquely determined and define the power order and exponential order of Ω , respectively. A simple example of an (α, ω)-weight function is furnished by Ω (t) = (1 + βt)α eωt

(t ∈ R+ )

A generalization of the Widder–Arendt theorem

3

with α, β ∈ R+ and ω ∈ R. It is readily verified that if Ω is an (α, ω)-weight function, then ω is a bound for L1 (R+ , Ω ). Given two normed vector spaces E and F , denote by L(E, F ) the space of all bounded linear operators from E into F . Recently, J. Kisy´ nski [17] established the following result: Theorem 1.1 (Widder–Arendt–Kisy´ nski). Let Ω be an (α, ω)-weight function on R+ , let E be a Banach space, and let r : (ω, ∞) → E be a function. Then r can be represented in the form r(λ) = T (−λ ) (λ > ω) ∞ for some T ∈ L(L1 (R+ , Ω ), E) if and only if r ∈ CW ((ω, ∞), E; Ω ). If there exists a T ∈ L(L1 (R+ , Ω ), E) such that r(λ) = T (−λ ) for all λ > ω, then T is unique and kT k = krkW,Ω,ω .

This theorem is a generalization of the Widder–Arendt representation theorem from Laplace transform theory. D. V. Widder established Theorem 1.1 in the case that E = F and Ω is of the form Ω (t) = eωt (t ∈ R+ , ω ∈ R) (cf. [21, pp. 315–316] and [22, p. 157]). Since L(L1 (R+ , Ω ), F) is isometrically isomorphic to the space L∞ (R+ , Ω −1 ) of all (equivalence classes of) F-valued Lebesgue measurable functions f on R+ for which f Ω −1 is essentially bounded, Theorem 1.1 in that case can be viewed as a characterization of the Laplace transforms of elements of L∞ (R+ , Ω −1 ). Extension to the case in which E is not necessarily F and Ω is still of the form Ω (t) = eωt (t ∈ R+ , ω ∈ R) is due to W. Arendt [1]. Arendt’s approach, drawing on Widder’s result, relies on reduction of the vector case to the scalar one. A direct proof of Arendt’s result was given by B. Hennig and F. Neubrander [14] (see also [18]). Another proof was offered by A. Bobrowski [3]. R. deLaubenfels, Z. Huang, S. Wang and Y. Wang [7] proved a special case of Theorem 1.1 in which Ω (t) = (1 + t)k (t ∈ R+ , k ∈ Z+ ). Kisy´ nski established a simultaneous generalization of the result of Arendt and that of deLaubenfels et al.. The aim of this paper is establish a generalization of Theorem 1.1. In this generalization, formulated as Theorem 1.2 below, L1 (R+ , Ω ) is replaced by a more general L1 space, not necessarily a weighted convolution algebra. This space may in particular be of the form L1 (R+ , Ω ), where Ω is a weight function with a bound, e.g., an (α, ω)-weight function. A similar result was recently, independently obtained by Bobrowski [4]. Let m be a non-negative Borel measure on R+ . Let L1 (R+ , m) be the space of all (classes of) F-valued m-integrable functions on R+ . A bound for m is an element ω of {−∞} ∪ R such that −µ ∈ L1 (R+ , m) for each µ > ω. If ω is a bound for m, then αk,µ ∈ L1 (R+ , m) (and hence also βk,µ ∈ L1 (R+ , m) and γk,µ ∈ L1 (R+ , m)) for all k ∈ Z+ and all µ > ω. Note that if a measure m admits a bound, then it is Radon; that is to say, m is finite on every bounded Borel subset of R+ . Let m be a non-negative Borel measure on R+ with bound ω. Let E be a Banach space. For r ∈ C ∞ ((ω, ∞), E), set krkW,m,ω = sup{kr(k) (λ)k/ kβk,λ k1,m | k ∈ Z+ , λ ∈ (ω, ∞)}

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∞ and define the Widder space CW ((ω, ∞), E; m) by ∞ CW ((ω, ∞), E; m) = {r ∈ C ∞ ((ω, ∞), E) | krkW,m,ω < ∞}. ∞ Endowed with the norm k · kW,m,ω , CW ((ω, ∞), E; m) is a Banach space. The main result of this paper is the following theorem:

Theorem 1.2. Let m be a non-negative Borel measure on R+ with bound ω, let E be a Banach space, and let r : (ω, ∞) → E be a function. Then r can be represented in the form r(λ) = T (−λ ) (λ > ω) ∞ for some T ∈ L(L1 (R+ , m), E) if and only if r ∈ CW ((ω, ∞), E; m). If there exists a T ∈ L(L1 (R+ , m), E) such that r(λ) = T (−λ ) for all λ > ω, then T is unique and kT k = krkW,m,ω .

As is the case with all versions of the Widder–Arendt theorem, the proof of Theorem 1.2 is based on an approximation argument. The method we use draws on the technique applied by Kisy´ nski in [16] to prove the classical Widder–Arendt theorem. Bobrowski proved a result similar to Theorem 1.2 employing the same technique with which he had earlier re-established the classical Widder–Arendt theorem (cf. [2]) and which goes back to Phillips [19] (see also [15, Theorem 6.6.3]). The remainder of the paper is organized as follows. Section 2 collects together a number of technical results. Section 3 is devoted to the proof of Theorem 1.2. In §4 the Widder spaces characterized in Theorem 1.2 are identified as spaces of the Laplace–Stieltjes transforms of certain vector-valued measures. Finally, §5 gives a corollary to Theorem 1.2 that characterizes pseudo-resolvents with values in a Banach algebra, satisfying a growth condition of Hille–Yosida type. Being of significance for the theory of semigroups of operators, the latter result provides motivation for conceiving Theorem 1.2 in the first place. 2. Auxiliary results We begin by establishing a number of technical results that will form a basis for the proof of Theorem 1.2. Denote by k · k∞ the uniform norm kf k∞ = supt∈R+ |f (t)|. For each λ ∈ R, let Cb (R+ , λ ) be the space of all F-valued continuous functions on R+ such that λ f is bounded. Under the norm kf k∞,λ = kλ f k∞ , Cb (R+ , λ ) is a Banach space. Note that −µ ∈ Cb (R+ , λ ) for each µ ≥ λ, and also αk,µ ∈ Cb (R+ , λ ) for each k ∈ N and each µ > λ. The following result is the key to all what ensues next. It is a generalization of a wellknown result from the theory of Borel summability [15, Equation (10.4.13)]. Its proof is based on a refinement of a well-known probabilistic argument [12, Chapter VII, Lemma 1 and Equation (1.5)].

A generalization of the Widder–Arendt theorem

5

Proposition 2.1. Let λ ∈ R and let f ∈ Cb (R+ , λ ). Then, for each a > 0, the series   ∞ X k f γk,a (t) a k=0

converges locally uniformly in t ∈ R+ . Furthermore, for each λ0 < λ, there exists a0 > 0 such that ∞   X f k kγk,a k sup (2.1) ∞,λ0 ≤ kf k∞,λ . a a>a0 k=0

We also have

  ∞ X k f (t) = lim f γk,a (t) a→∞ a

(2.2)

k=0

for all t ∈ R+ . Proof . Since, for each t ∈ R+ , |f (t)| ≤ kf k∞,λ e−λt ,

(2.3)

∞   ∞ k X X −λk/a (at) f k γk,a (t) ≤ kf k∞,λ e e−at a k!

(2.4)

we have

k=0

k=0

= kf k∞,λ exp[at(e−λ/a − 1)] for all a > 0 and all t ∈ R+ . This immediately implies that, for each a > 0, the series P∞ k=0 f (k/a) γk,a (t) converges locally uniformly in t ∈ R+ . Note that lim a(1 − e−λ/a ) = λ. (2.5) a→∞

0

0

Hence if λ satisfies λ < λ, then there exists a0 > 0 such that a(e−λ/a − 1) < −λ0 for all a > a0 . Consequently, in view of (2.4), ∞   X f k γk,a (t) ≤ kf k∞,λ e−λ0 t a k=0

0

for all a > a and all t ∈ R+ , which gives (2.1). It remains to prove (2.2). Fix t ∈ R+ arbitrarily. Given a > 0, let Yat be a Poisson random variable with parameter at, carried by a probability space (Ω, M, P); that is to say, Yat is a Z+ -valued random variable such that P [Yat = k] =

(at)k −at e = γk,a (t) k!

for all k ∈ Z+ . Set Xa,t = Yat /a. Denoting by E [X] the expected value of the random variable X, we clearly have   ∞ X k f E [f (Xa,t )] = γk,a (t) a k=0

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and further E [f (Xa,t ) − f (t)] =

  ∞ X k f γk,a (t) − f (t). a

k=0

Thus, in order to prove (2.2), it suffices to show that lim E [f (Xa,t ) − f (t)] = 0.

a→∞

(2.6)

Choose  > 0 arbitrarily. Since f is continuous, there exists δ > 0 such that |f (x) − f (t)| < /2 for every x ∈ R+ with |x − t| < δ. Let Aa,t,δ = {ω ∈ Ω | |Xa,t (ω) − t| < δ}. As is customary, given a set A, denote by 1A the characteristic function of A. Now     |E [f (Xa,t ) − f (t)] | ≤ E |f (Xa,t ) − f (t)|1Aa,t,δ + E |f (Xa,t ) − f (t)|1Ω\Aa,t,δ . (2.7) Of course, |f (Xa,t ) − f (t)|1Aa,t,δ ≤ (/2)1Aa,t,δ , and so   E |f (Xa,t ) − f (t)|1Aa,t,δ ≤ (/2)P [Aa,t,δ ] ≤ /2.

(2.8)

We shall prove shortly that   lim E |f (Xa,t ) − f (t)|1Ω\Aa,t,δ = 0.

a→∞

(2.9)

Assuming this for now, select a0 > 0 so that   E |f (Xa,t ) − f (t)|1Ω\Aa,t,δ < /2 for all a > a0 . Then, on account of (2.7) and (2.8), E [|f (Xa,t ) − f (t)|] <  for all a > a0 . We thus see that (2.6) holds. We proceed to prove (2.9). By the Cauchy–Schwarz inequality,       E |f (Xa,t ) − f (t)|1Ω\Aa,t,δ ≤ (E |f (Xa,t ) − f (t)|2 )1/2 (E (1Ω\Aa,t,δ )2 )1/2 (2.10)   = (E |f (Xa,t ) − f (t)|2 )1/2 (P [Ω \ Aa,t,δ ])1/2 . By the triangle inequality for the L2 norm,       (E |f (Xa,t ) − f (t)|2 )1/2 ≤ (E |f (Xa,t )|2 )1/2 + (E |f (t)|2 )1/2   = (E |f (Xa,t )|2 )1/2 + |f (t)|. Clearly,

(2.11)

∞   2   X f k γk,a (t) E |f (Xa,t )|2 = a k=0

and, by (2.3), ∞ ∞   2 X X f k γk,a (t) ≤ kf k2∞,λ e−2λk/a γk,a (t) = kf k2∞,λ exp[at(e−2λ/a − 1)]. a k=0

k=0

A generalization of the Widder–Arendt theorem

7

Since lima→∞ a(e−2λ/a − 1) = −2λ, it follows that   lim sup E |f (Xa,t )|2 ≤ kf k2∞,λ e−2λt a→∞

and further, by (2.11),   lim sup(E |f (Xa,t ) − f (t)|2 )1/2 ≤ kf k∞,λ e−λt + |f (t)|.

(2.12)

a→∞

On the other hand, since Ω \ Aa,t,δ = {ω ∈ Ω | |Xa,t (ω) − a| ≥ δ}, an application of Chebyshev’s inequality implies that   E (Xa,t − t)2 P [Ω \ Aa,t,δ ] ≤ . δ2

(2.13)

It is a well-known property of Poisson variables that E [Yat ] = var [Yat ] = at, where var [X]  denotes the  variance of the random variable X, defined—let us recall—by 2 var [X] = E |X − E [X] | . Hence E [Xa,t ] = and var [Xa,t ] =

E [Yat ] =t a var [Yat ] t = . a2 a

Rewriting the last equality as   t E (Xa,t − t)2 = a and combining it with (2.13), we conclude that lim P [Ω \ Aa,t,δ ] = 0.

a→∞

This together with (2.10) and (2.12) implies (2.9).



For each λ ∈ R, let P (R+ , λ) be the linear space spanned by the set {−µ | µ > λ}. Clearly, [ P (R+ , λ) = P (R+ , ν). (2.14) ν>λ

Under pointwise multiplication, P (R+ , 0) is an algebra. Moreover, all the P (R+ , λ) (λ ∈ R) are modules for P (R+ , 0). Let C0 (R+ ) be the space of all F-valued continuous functions f on R+ vanishing at infinity.

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Proposition 2.2. Let m be a non-negative Borel measure on R+ with bound ω. Then, for each µ > ω, P (R+ , µ) is dense in L1 (R+ , m). In particular, P (R+ , ω) is dense in L1 (R+ , m). Proof . It suffices to prove the first assertion, the other being evident in light of (2.14). Fix µ > ω arbitrarily. Let n be the Borel measure on R+ defined by dn = −µ dm. Clearly, n is finite. Let f ∈ L1 (R+ , m). Then µ f ∈ L1 (R+ , n) and, given  > 0, there exists ϕ ∈ C0 (R+ ) such that kµ f − ϕk1,n < /2. A standard argument based on the Stone– Weierstrass theorem shows that the algebra P (R+ , 0) is dense in C0 (R+ ). Therefore, there exists p ∈ P (R+ , 0) such that kϕ − pk∞ < /(2 k−µ k1,m ). Since kf − −µ pk1,m ≤ kf − −µ ϕk1,m + k−µ ϕ − −µ pk1,m , kf − −µ ϕk1,m = kµ f − ϕk1,n , k−µ ϕ − −µ pk1,m ≤ k−µ k1,m kϕ − pk∞ , it follows that kf − −µ pk1,m < . Noting that −µ p ∈ P (R+ , µ) finishes the proof.



Proposition 2.3. Let m be a non-negative Borel measure on R+ with bound ω, let λ > ω, and let f ∈ Cb (R+ , λ ). Then ∞   X f k kγk,a k (2.15) lim 1,m = kf k1,m . a→∞ a k=0

Proof . Applying the first assertion in Proposition 2.1 with |f | in place of f , we deduce that, for each a > 0, the series ∞   X f k γk,a (t) (t ∈ R+ ) a k=0

defines a continuous function. Denote this function by ψa . Select λ0 so that ω < λ0 < λ. The second assertion in Proposition 2.1 ensures that there exists a0 > 0 such that kψa k∞,λ0 ≤ kf k∞,λ for all a > a0 . Therefore, for each a > a0 , ψa is a function in Cb (R+ , λ0 ) and ψa ≤ kf k∞,λ −λ0 . To prove (2.15), it suffices to show that  ∞  X k lim f an kγk,an k1,m = kf k1,m n→∞

(2.16)

(2.17)

k=0

for any sequence {an } in (a0 , ∞) diverging to infinity. Let {an } be a sequence in (a0 , ∞) with limn→∞ an = ∞. The third assertion in Proposition 2.1 applied to |f | yields lim ψan (t) = |f (t)|

n→∞

A generalization of the Widder–Arendt theorem

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for all t ∈ R+ . As λ0 > ω, the function −λ0 is m-integrable, and so, in view of (2.16), {ψan } has an m-integrable majorant. By Lebesgue’s dominated convergence theorem, Z Z lim ψan (t) dm(t) = |f (t)| dm(t) = kf k1,m . n→∞

R+

R+

On the other hand, an application of Levi’s monotone convergence theorem implies that   Z ∞ Z X f k γk,a (t) dm(t) ψa (t) dm(t) = a R+ k=0 R+ ∞   X f k kγk,a k = 1,m a k=0

for all a > 0. Thus (2.17) is established.



Proposition 2.4. Let m be a non-negative Borel measure on R+ with bound ω, and ∞ let E be a Banach space. Then any function in CW ((ω, ∞), E; m) is real analytic. ∞ Proof . Let r ∈ CW ((ω, ∞), E; m) and let  > 0. If ξ > ω + 2, then ∞ X (t)k k=0

k!

e−ξt = e−(ξ−)t ≤ e−(ω+)t

for all t ∈ R+ . Integrating with respect to the variable t against m, we obtain ∞ X

k kαk,ξ k1,m ≤ −(ω+) 1,m .

k=0

Hence, for each k ∈ Z+ ,

kαk,ξ k1,m ≤ −(ω+) 1,m −k

and further

1 (k) kr (ξ)k ≤ krkW,m,ω −(ω+) 1,m −k . k! Fix λ > ω + 2 arbitrarily. Choose δ > 0 so that λ − δ > ω + 2. If x ∈ (λ − δ, λ + δ), then, for each ξ belonging to the interval joining λ and x,

1 (k) kr (ξ)k |x − λ|k ≤ krkW,m,ω −(ω+) 1,m (|x − λ|/)k . k! If |x − λ| < −1 , Taylor’s formula implies that ∞ X 1 (k) r (λ)(x − λ)k k!

k=0

converges to r(x). Thus r is real analytic at λ, and hence is real analytic on all of (ω + 2, ∞). Since  is arbitrary, the theorem follows. 

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Proposition 2.5. Let m be a non-negative Borel measure on R+ with bound ω, let ∞ E be a Banach space, and let r ∈ CW ((ω, ∞), E; m). If λ > ω, then there exists a1 > 0 such that ∞ X (−1)k k −λk/a (k) a e r (a) = r(a(1 − e−λ/a )) (2.18) k! k=0

for all a > a1 . Proof . Let λ > ω. Since −λ ∈ Cb (R+ , λ ), it follows from Proposition 2.3 that there exists a0 > 0 such that ∞ X e−λk/a kγk,a k1,m < ∞ (2.19) k=0

for all a > a0 . Taking into account (2.5), we infer that there exists a00 > 0 such that a(1 − e−λ/a ) > ω

(2.20)

for all a > a00 . Let a1 be a positive number no smaller than both a0 and a00 . Fix a > a1 arbitrarily. Since 1 k (k) a kr (a)k ≤ krkW,m,ω kγk,a k1,m , k! if follows from (2.19) that ∞ X 1 k −λk/a (k) a e kr (a)k < ∞ k!

(2.21)

k=0

for all a > a1 . Thus the power series ∞ X 1 (k) r (a)tk k!

k=0 −λ/a

converges for all |t| ≤ ae . Denote by ϕ the function defined by this series. Clearly, ϕ is real analytic on I = (−ae−λ/a , ae−λ/a ). We claim that ϕ is continuous on I = [−ae−λ/a , ae−λ/a ]. It suffices to prove that ϕ is continuous at each endpoint of I. Let σ ∈ {−1, 1}. Suppose that t satisfies |t| ≤ ae−λ/a and is so close to σae−λ/a that |σae−λ/a − t| < ae−λ/a /2. Then |ae−λ/a − σt| < ae−λ/a /2, and so σt > ae−λ/a /2, showing that σt is positive and equal to |t|. Now |(σtae−λ/a )k − tk | = |(ae−λ/a )k − (σt)k | = (ae−λ/a )k − |t|k for all k ∈ Z+ . Consequently, kϕ(σae−λ/a ) − ϕ(t)k ≤ =

∞ X |(σtae−λ/a )k − tk | k=0 ∞ X k=0

k!

kr(k) (a)k ∞

X |t|k (ae−λ/a )k (k) kr (a)k − kr(k) (a)k. k! k! k=0

A generalization of the Widder–Arendt theorem

11

By (2.21) and Abel’s limit theorem (cf. [20, Chapter III, §2, Theorem 2.1]), or alternatively by (2.21) and Levi’s monotone convergence theorem applied to the counting measure on Z+ , the right hand side above converges to zero as |t| → ae−λ/a . This implies that ϕ is continuous at σae−λ/a , establishing the claim. In light of (2.20), the function ψ(t) = r(a + t) is well defined for t ∈ I. By Proposition 2.4, ψ is real analytic on I, and, clearly, has the same derivatives at the value t = 0 as ϕ. Therefore ϕ = ψ on I. Since both ϕ and ψ are continuous on I, they coincide on I, in particular at the value t = −ae−λ/a . This establishes (2.18).  Proposition 2.6. Let m be a non-negative Borel measure on R+ with bound ω. Then the function (ω, ∞) 3 λ 7→ −λ ∈ L1 (R+ , m) is infinitely differentiable in the norm topology of L1 (R+ , m), and dk −λ = (−1)k βk,λ dλk for each k ∈ Z+ and each λ > ω. Proof . Given that β0,λ = −λ , it suffices to show that, for each k ∈ Z+ , the function (ω, ∞) 3 λ 7→ βk,λ ∈ L1 (R+ , m) is differentiable in the norm topology of L1 (R+ , m) and dβk,λ = −βk+1,λ . dλ Fix k ∈ Z+ arbitrarily. Direct computation shows that ∂βk,λ (t) = −βk+1,λ (t) ∂λ

(2.22)

for all λ, t ∈ R. Fix λ > ω arbitrarily, and next select ω 0 and ω 00 so that ω < ω 0 < ω 00 < λ. Let h be such that 0 < |h| < ω 00 − ω 0 . Taking into account (2.22) and Taylor’s formula, we see that, given t ∈ R+ , there exists ξt in the interval joining λ and λ + h such that βk,λ+h (t) − βk,λ (t) = −βk+1,ξt (t). h Clearly, βk+1,λ ∈ Cb (R+ , ω 00 ) and 00

tk+1 e−λt ≤ kβk+1,λ k∞,ω00 e−ω t . Hence 00

βk+1,ξt (t) = tk+1 e−ξt t ≤ tk+1 e−λt e|h|t ≤ kβk+1,λ k∞,ω00 e−ω t e(ω 0

= kβk+1,λ k∞,ω00 e−ω t , showing that the functions t 7→

βk,λ+h (t) − βk,λ (t) h

(0 < |h| < ω 00 − ω 0 )

00

−ω 0 )t

12

W. Chojnacki

are dominated by an m-integrable function. Let {hn } be an arbitrary sequence in (0, ω 00 − ω 0 ) tending to zero as n → ∞. Since, by (2.22), the sequence {(βk,λ+hn − βk,λ )/hn } tends pointwise to −βk+1,λ as n → ∞, an appeal to Lebesgue’s dominated convergence theorem reveals that

βk,λ+hn − βk,λ

lim + β = 0, k+1,λ

n→∞ hn 1,m which is the desired result.



3. Proof of the main result Proof of Theorem 1.2. Let r : (ω, ∞) → E be a function such that r(λ) = T (−λ ) for some bounded linear operator T : L1 (R+ , m) → E and all λ > ω. Since, according to Proposition 2.2, P (R+ , ω) is dense in L1 (R+ , m), T is uniquely determined by its values taken on at the −λ (λ > ω). By Proposition 2.6, the function (ω, ∞) 3 λ 7→ −λ ∈ L1 (R+ , m) is infinitely differentiable in the norm topology of L1 (R+ , m), and dk −λ = (−1)k βk,λ dλk for each k ∈ Z+ and each λ > ω. Therefore r is infinitely differentiable and T (βk,λ ) = (−1)k r(k) (λ) for each k ∈ Z+ and each λ > ω. Consequently, we have krkW,m,ω ≤ kT k, and in particular krkW,m,ω is finite. Now—as a moment’s reflection reveals—to complete the ∞ proof, it suffices to show that if r is a function in CW ((ω, ∞), E; m), then there exists 1 a bounded linear operator T : L (R+ , m) → E such that T (−λ ) = r(λ) for each λ > ω and kT k ≤ krkW,m,ω . ∞ Suppose then that r ∈ CW ((ω, ∞), E; m). Define a linear operator T : P (R+ , ω) → E as follows: for each f ∈ P (R+ , ω) representable as X f= aλ −λ (aλ ∈ F) λ∈Λ

where Λ is a finite subset of (ω, ∞), let T (f ) =

X

aλ r(λ).

λ∈Λ

Clearly, T (−λ ) = rλ for all λ > ω. We shall prove that kT (f )k ≤ krkW,m,ω kf k1,m .

(3.1)

This will guarantee that T is well-defined and also that T is bounded with kT k ≤ krkW,m,ω provided that P (R+ , ω) is considered with the norm k · k1,m . Since P (R+ , ω) is dense in L1 (R+ , m), T can next be extended by continuity to a bounded linear operator from L1 (R+ , m) to E, the norm of the extension being equal to kT k.

A generalization of the Widder–Arendt theorem

13

In order to prove (3.1), we introduce, for each n ∈ N and each a > 0, a linear operator Tn,a : P (R+ , ω) → E defined by Tn,a (f ) =

n X (−1)k

  k a f r(k) (a) k! a

k=0

k

(f ∈ P (R+ , ω)).

P Fix f = λ∈Λ aλ −λ arbitrarily and choose µ > ω so that Λ ⊂ (µ, ∞). Clearly, −λ ∈ Cb (R+ , µ ) for all λ ∈ Λ. By Proposition 2.5, there exists a1 > 0 such that lim Tn,a (−λ ) = r(a(1 − e−λ/a ))

n→∞

for all λ ∈ Λ and all a > a1 . Hence lim Tn,a (f ) =

n→∞

X

aλ r(a(1 − e−λ/a ))

λ∈Λ

for all a > a1 . Since r is continuous, it follows from (2.5) that lim r(a(1 − e−λ/a )) = r(λ).

a→∞

Consequently, h i lim lim T (f ) = T (f ). n,a a→∞ a>a1

n→∞

(3.2)

Since, for each n ∈ N and each a > 0,   n   n X X 1 k k (k) f k kγk,a k f a kr (a)k ≤ krkW,m,ω kTn,a (f )k ≤ 1,m k! a a k=0 k=0 ∞   X f k kγk,a k , ≤ krkW,m,ω 1,m a k=0

invoking Proposition 2.3 we conclude that lim sup [lim sup kTn,a (f )k] ≤ krkW,m,ω kf k1,m . a→∞

n→∞

This together with (3.2) implies (3.1).



4. A link with the Laplace–Stieltjes transform Let m be a non-negative Borel measure on R+ , and let E be a Banach space. Here we ∞ identify Widder spaces of the form CW ((ω, ∞), E; m), where ω is a bound for m, as spaces of the Laplace–Stieltjes transforms of certain E-valued measures. This characterization is based on the fact that the action of operators in L(L1 (R+ , m), E) can be expressed in terms of integrals with respect to E-valued measures. We begin by recalling that a ring, or a clan, of subsets of a set X is a non-empty collection of subsets of X that is closed under pairwise unions and relative complementation

14

W. Chojnacki

(cf. [9, p. 1], [13, p. 19]). Let R(R+ , m) be the ring of m-measurable subsets A of R+ such that m(A) < ∞. Note that R(R+ , m) is not an algebra of sets unless m(R+ ) < ∞. Let M (R+ , E; m) be the space of all E-valued measures µ on R(R+ , m) for which there exists a non-negative C = C(µ) such that kµ(A)k ≤ Cm(A) for all A ∈ R(R+ , m). The σ-additivity of m immediately implies that every measure in M (R+ , E; m) is σ-additive. Equipped with the norm kµk∞,m = sup{C ∈ R+ | kµ(A)k ≤ Cm(A) for all A ∈ R(R+ , m)} = sup{kµ(A)k/m(A) | A ∈ R(R+ , m) with m(A) > 0}, M (R+ , E; m) is a Banach space. As it turns out, the spaces L(L1 (R+ , m), E) and M (R+ , E; m) are isometrically isomorphic. Below we outline the proof of this result. Let T ∈ L(L1 (R+ , m), E) be a continuous linear operator. For each A ∈ R(R+ , m), define µT (A) to be T (1A ). The set function µT : R(R+ , m) → E, A 7→ µT (A), is easily seen to be a measure in M (R+ , E; m) satisfying kµT k∞,m ≤ kT k. Accordingly, the mapping L(L1 (R+ , m), E) 3 T 7→ µT ∈ M (R+ , E; m) is a linear contraction. We proceed to define an inverse map. Let S(R+ , m) be the set of all simple functions from R+ into F Pn of the form i=1 ai 1Ai , where ai ∈ F and Ai ∈ R(R+ , m) for each i = 1, . . . , n. Clearly, S(R+ , m) is a dense linear subspace of L1 (R+ , m). Fix µ ∈ M (R+ , E; m) arbitrarily. For each f ∈ S(R+ , m), let X Tµ (f ) = a µ(f −1 ({a})). a∈F

A routine verification shows that S(R+ , m) 3 f 7→ Tµ (f ) ∈ E is a continuous linear operator with norm not greater than kµk∞,m . This operator can further be extended by continuity to a continuous linear operator Tµ : L1 (R+ , m) → E with norm not greater than kµk∞,m . Clearly, M (R+ , E; m) 3 µ 7→ Tµ ∈ L(L1 (R+ , m), E) is a linear contraction. It is readily verified that the mappings T 7→ µT and µ 7→ Tµ are mutually inverse and as such determine an isometric isomorphism between L(L1 (R+ , m), E) and M (R+ , E; m). Incidentally, note that if f ∈ L1 (R+ , m) and µ ∈ M (R+ , E; m), then the element Tµ (f ) of E coincides with theRintegral of f with respect to µ (see [8, pp. 5–6]). Therefore Tµ (f ) can also be written as R+ f dµ. The space M (R+ , E; m) can be further characterized for E having the Radon–Nikodym property or RNP. Recall that a Banach space E has the RNP provided that if ν is an E-valued measure of finite variation and ν is absolutely continuous with respect to a finite R scalar measure n, then there is an E-valued measurable function g so that ν(A) = A g dn for every measurable set A, the integral being taken in the sense of Bochner. An equivalent requirement is that every Lipschitz function from R into E be differentiable almost everywhere with respect to Lebesgue measure. The RNP is a hereditary property (i.e., passes to closed subspaces) and is enjoyed (amongst others) by any reflexive space, any separable dual space, and any `1 (Γ) space, where Γ is a set (see [8, Chapter III]). Let L∞ (R+ , E; m) be the space of (equivalence classes of) E-valued m-measurable essentially bounded functions on R+ , with the norm kf k∞,m = inf{sup{kf (t)k | t ∈ A} | A ⊂ R+ m-measurable with m(R+ \ A) = 0}.

A generalization of the Widder–Arendt theorem

15

If E has the RNP, then the spaces L∞ (R+ , E; m) and M (R+ , E; m) are isometrically isomorphic under the correspondence g ↔ µ, where g ∈ L∞ (R+ , E; m) and µ ∈ M (R+ , E; m), defined by Z µ(B) = g dm (B ∈ R(R+ , m)), B

where the integral is taken in the sense of Bochner (see [8, Chapter III]). Consequently, if (g, µ) is a pair of corresponding elements of L∞ (R+ , E; m) and M (R+ , E; m), then any integral of the form Z (f ∈ L1 (R+ , m))

f dµ R+

can be represented as the Bochner integral Z Z f dµ = R+

f g dm.

R+

In light of the comments above, we can now formulate two immediate corollaries to ∞ Theorem 1.2. The first identifies any CW ((ω, ∞), E; m), where ω is a bound for m, as the space of the Laplace–Stieltjes transforms on (ω, ∞) of measures in M (R+ , E; m), whereas ∞ the other, under the assumption that E has the RNP, identifies any CW ((ω, ∞), E; m) as the space of the Laplace–Stieltjes transforms on (ω, ∞) of E-valued measures on R(R+ , m) absolutely continuous with respect to m with density in L∞ (R+ , E; m). Theorem 4.1. Let m be a non-negative Borel measure on R+ with bound ω, let E be a Banach space, and let r : (ω, ∞) → E be a function. Then r can be represented in the form Z r(λ) =

−λ dµ (λ > ω) R+

∞ for some µ ∈ M (R+ , E; m) if and R only if r ∈ CW ((ω, ∞), E; m). If there exists a µ ∈ M (R+ , E; m) such that r(λ) = R+ −λ dµ for all λ > ω, then µ is unique and kµk∞,m = krkW,m,ω .

Theorem 4.2. Let m be a non-negative Borel measure on R+ with bound ω, let E be a Banach space with the RNP, and let r : (ω, ∞) → E be a function. Then r can be represented in the form Z r(λ) =

−λ g dm

(λ > ω)

R+ ∞ for some g ∈ L∞ (R+ , E; m) if and R only if r ∈ CW ((ω, ∞), E; m). If there exists a g ∈ ∞ L (R+ , E; m) such that r(λ) = R+ −λ g dm for all λ > ω, then g is unique and kgk∞,m = krkW,m,ω .

One more comment is in order. If µ is a measure in M (R+ , E; m), where E is a Banach space and m is a non-negative Borel measure on R+ with bound ω, then, for

16

W. Chojnacki

R every λ > max{ω, 0}, the Laplace–Stieltjes transform R+ −λ dµ can be represented as the Laplace–Carson transform, or λ-multiplied Laplace transform Z Z ∞ −λ dµ = λ e−λt gµ (t) dt, (4.1) 0

R+

where gµ is the function from R+ into E given by gµ (t) = µ([0, t]) for all t ∈ R+ , and the right-hand side integral is taken in the sense of Bochner. To prove this assertion, define f : R+ → L1 (R+ , m) by f (t) = 1[0,t] for all t ∈ R+ . We shall show that f is (strongly) Borel measurable. Let A = {t ∈ R+ : m({t}) > 0}. Since m is finite on every bounded Borel subset of R+ , A is at most countable. Let A = {an : n ∈ N} be an enumeration of A. Write f as f = f1 + f2 , where f1 and f2 are the functions from R+ into L1 (R+ , m) defined by f1 (t) = 1[0,t]∩A and f2 (t) = 1[0,t]∩(R+ \A) for all t ∈ R+ . Given n ∈ N, define hn : R+ → L1 (R+ , m) by hn (t) = 1[0,t]∩{an } for all t ∈ R+ . Each hn is Borel measurable, being equal to zero on [0, an ) and to 1{an } on [an , ∞). It is readily P seen that, for each t ∈ R+ , f1 (t) = n∈N hn (t), the series being absolutely convergent in the norm topology of L1 (R+ , m). This implies that f1 is Borel measurable. On the other hand, since m restricted to R+ \ A is continuous, it follows that f2 is continuous and hence Borel measurable. Thus f is Borel measurable. As an immediate consequence, we see that, for every λ ∈ R, the function R+ 3 t 7→ λe−λt 1[0,t] ∈ L1 (R+ , m) is Borel measurable. Now, for each λ > 0 and each s ∈ R+ , Z ∞ Z ∞ λ e−λt 1[0,t] (s) dt = λ e−λt dt = e−λs (4.2) 0

s

and further, for every λ > max{ω, 0}, Z λ

∞

e 0

−λt



1[0,t] dt = λ 1,m

Z

∞

e

−λt

"Z

1[0,t] (s) dm(s) dt

0

R+

 Z λ

Z = R+

Z

#

∞

e

−λt

 1[0,t] (s) dt dm(s)

0

e−λs dm(s) < +∞

= R+

by Fubini’s theorem. Fix λ > max{ω, 0} arbitrarily. According to Bochner’s integrability criterion (cf. [23, Chapter 5, §5, Theorem 1]), the function R+ 3 t 7→ λe−λt 1[0,t] ∈ L1 (R+ , m) is Bochner integrable with respect to Lebesgue measure on R+ . In view of R ∞ −λt (4.2), the Bochner integral 0 λe 1[0,t] dt is the element of L1 (R+ , m) represented by −λ . Now, by the continuity of Tµ (as an operator from L1 (R+ , m) to E), the function R+ 3 t 7→ λe−λt Tµ (1[0,t] ) ∈ E is Bochner integrable and Z Tµ (−λ ) =

∞

λe−λt Tµ (1[0,t] ) dt.

0

Taking into account that gµ (t) = Tµ (1[0,t] ) for each t ∈ R+ , we finally obtain (4.1).

A generalization of the Widder–Arendt theorem

17

5. An application to pseudo-resolvents Here we give one application of Theorem 1.2 of importance for operator semigroup theory. Let A be a Banach algebra (with identity or not) and let U be a subset of F. A function r : U → A, λ 7→ rλ , is called a pseudo-resolvent if it satisfies the following Hilbert equation: rλ − rµ = (µ − λ)rλ rµ (λ, µ ∈ U ). Let U be an open subset of R and let r : U → A be a pseudo-resolvent. It is well known that r is then infinitely differentiable, in fact real analytic, and dk rλ = (−1)k k! rλk+1 dλk for all k ∈ Z+ (cf. [11, Chapter IX, §1], [15, §5.8], [23, Chapter VIII, §4]). Thus, if m is a non-negative Borel measure on R+ with bound ω and U = (ω, ∞), then krkW,m,ω = sup{krλk+1 k/ kαk,λ k1,m | k ∈ Z+ , λ ∈ (ω, ∞)}. When m has a density Ω (with respect to Lebesgue measure) of the form Ω (t) = eωt (t ∈ R+ , ω ∈ R), we have kαk,λ k1,m = (λ − ω)−(k+1) for all k ∈ Z+ and all λ > ω, and the equality above reduces to krkW,Ω,ω = sup{(λ − ω)k krλk k | k ∈ N, λ ∈ (ω, ∞)}. Now observe that krkW,Ω,ω < ∞ is nothing else but the Hille–Yosida condition intervening in the Hille–Yosida theorem on the generation of one-parameter semigroups of operators (cf. [10, Chapter VIII, §1, Theorem 13], [15, Theorem 12.3.1], [23, Chapter 9, §7]). By analogy, when m is a non-negative Borel measure on R+ with bound ω and r : (ω, ∞) → A is a pseudo-resolvent, r will be said to satisfy the Hille–Yosida condition if krkW,m,ω < ∞. Direct verification shows that if Ω is a weight function on R+ with bound ω, then  : (ω, ∞) → L1 (R+ , Ω ), λ 7→ −λ , is a pseudo-resolvent with kkW,Ω,ω = 1. As will become clear in a moment, from a certain point of view  can be regarded as a canonical pseudo-resolvent in L1 (R+ , Ω ). Let Ω be a weight function on R+ with bound ω, suppose that r : (ω, ∞) → A, λ 7→ rλ , is a pseudo-resolvent, and let T : L1 (R+ , Ω ) → A be a bounded linear operator such that rλ = T (−λ ) for all λ > ω. Then, for all λ, µ ∈ (ω, ∞), (µ − λ)T (−λ ∗ −µ ) = T (−λ − −µ ) = T (−λ ) − T (−µ ) because  is a pseudo-resolvent, and (µ − λ)T (−λ )T (−µ ) = (µ − λ)rλ rµ = rλ − rµ = T (−λ ) − T (−µ ) because r is a pseudo-resolvent. Hence T (−λ ∗ −µ ) = T (−λ )T (−µ ) for all λ, µ ∈ (ω, ∞) with λ 6= µ. This identity readily extends to the case λ = µ, since the mapping

18

W. Chojnacki

(ω, ∞) 3 λ 7→ −λ ∈ L1 (R+ , Ω ) is continuous. Accordingly, T (f ∗ g) = T (f )T (g) for all f, g ∈ P (R+ , ω). Since P (R+ , ω) is dense in L1 (R+ , Ω ) and T is bounded, we have T (f ∗g) = T (f )T (g) for all f, g ∈ L1 (R+ , m). Thus T is a homomorphism from L1 (R+ , Ω ) to A. Coupled with Theorem 1.2, these observations lead to the following result: Theorem 5.1. Let A be a Banach algebra, let Ω be a weight function on R+ with bound ω, and let r : (ω, ∞) → A, λ 7→ rλ , be a pseudo-resolvent. Then r satisfies the Hille–Yosida condition krkW,Ω,ω = sup{krλk+1 k/ kαk,λ k1,Ω | k ∈ Z+ , λ ∈ (ω, ∞)} < ∞ if and only if there exists a bounded homomorphism T : L1 (R+ , Ω ) → A such that T (−λ ) = rλ for each λ ∈ (ω, ∞). Furthermore, if there exists a bounded homomorphism T : L1 (R+ , Ω ) → A such that T (−λ ) = rλ for each λ ∈ (ω, ∞), then T is unique and kT k = krkW,Ω,ω . In the case when Ω is an (α, ω)-weight function, this theorem is due to Kisy´ nski [17]. Related results can be found in [3, 5, 16]. The significance of Theorem 5.1 is that it can be used to develop a generalization of the Hille–Yosida theorem, and a generalization of the Trotter–Kato theorem on the convergence of sequences of one-parameter semigroups (cf. [23, Chapter 9, §12, Theorem 1]). Both these generalizations will involve semigroups {St }t∈R+ satisfying a growth condition of the form supt∈R+ (Ω (t))−1 kSt k < +∞, where Ω is a weight function on R+ . The interested reader is referred to [6, 16] for a formulation and validation of generalizations of the Hille–Yosida theorem and Trotter–Kato theorem based on the classical form of the Widder–Arendt theorem. In [17] a further generalization of the Hille–Yosida theorem is derived from a special case of Theorem 5.1 in which Ω is an (α, ω)-weight function. All of these generalizations of the Hille–Yosida theorem and Trotter–Kato theorem can eventually be subsumed by respective generalizations, derivable from Theorem 5.1, involving weight functions with a bound. A detailed exposition of these latter results will be presented elsewhere.

A generalization of the Widder–Arendt theorem

19

References 1. 2. 3.

4. 5. 6. 7.

8. 9. 10. 11. 12. 13. 14. 15. 16.

17. 18.

19. 20. 21. 22. 23.

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